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A powerful but simple matrix method for the digital computer manipulation of large sets of differential and algebraic equations is introduced to the field of power systems. The method is first illustrated with reference to the linearized equations representing a general-purpose turboalternator model. Subsequently, the form of the system equations obtained as a result of the matrix method is seen to provide a unified approach to the determination of system stability limits using Routh, Nyquist, or eigenvalue methods. The normal state space form of the system equations is also shown to facilitate control studies. A previously suggested performance criterion for an excitation system is generalized, and a systematic method for the simultaneous optimization of governor and exciter control loops is suggested. This method of optimization does not require the evaluation of system transient responses. The manner in which the normal state space form of the system equations leads naturally to the synthesis of controllers which are optimal with respect to a chosen scalar quadratic performance index is indicated. With this form of control, the structure is not defined a priori but emerges from the computational algorithm and it becomes unnecessary to specify the source or the magnitude of the stabilizing signals required to provide an improved dynamic performance. Finally, the matrix method is shown to be effective for the manipulation of the nonlinear machine equations and these are obtained in a new form which is directly amenable to digital or analog computer solution.